An Approach to the Log-Euclidean Mean via the Karcher Mean on Symmetric Cones

In a general symmetric cone, we show that certain sequence of the Karcher means converges to the Log-Euclidean mean by using the fact that the Karcher mean is the limit of inductive means. One can see this as a generalization of the Lie-Trotter formula of positive definite matrices into a symmetric cone setting via the least squares mean. 2010Mathematics Subject Classification. 47A64, 17C50, 15B48, 53C20. Key words and phrases. Lie-Trotter formula, Least squares mean, Symmetric cone, Hadamard space.